The Product of linear forms over function fields

The aim of this paper is to study the product of $n$ linear forms over function fields. We calculate the maximum value of the minima of the forms with determinant one when $n$ is small. The value is equal to the natural bound given by algebraic number theory. Our proof is based on a reduction theory of diagonal group orbits on homogeneous spaces. We also show that the forms defined algebraically correspond to periodic orbits with respect to the diagonal group actions.